A romp through the foothills of logic Session 3 › 2014 › 07 › fl3.pdf · Firstwe’vegottodrawthe emptytruth table.Toevaluatethat sequent you need a truth table. (Slide 10)

A romp through the foothills of logic – Session 3

It would be a good idea to watch the short podcast ‘Understanding Truth Tables’

before attempting this podcast.

Lecturer: (Slide 2) In the last session we learnt how to formalise English

arguments as sequents of the propositional calculus. Who can

tell me what a sequent is?

Male: Something in sequence – one, two, three, four, five, six,

seven?

Lecturer: No. What is a sequent?

[inaudible discussion]

Lecturer: No.

Male: Numerical order?

Lecturer: No. What is a sequent?

Male: A set of statements?

Male: Something used to make a statement?

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Lecturer: No, none of these things. It’s an argument claim.

There are semantic sequents and there are syntactic

sequents, and each of them is an argument claim.

The claim that it makes is of course the claim of any argument,

which is that ‘these premises entail this conclusion’.

Female: So is that a conclusion that you're talking about …?

Lecturer: No, I'm talking about the sequent, which is the argument claim,

which is the formalised claim that the premises entail the

conclusion.

Female: So that encompasses the whole thing?

Lecturer: Indeed. So we learnt how to formalise an English argument.

An English argument says that ‘this conclusion follows from

these premises’, and we formalise that as a sequent of the

propositional calculus, and each of the sequents tells us that

these premises entail this conclusion.

In other words, there’s no possible situation in which these

premises are true and this conclusion is false.

We saw that there were two types of sequents – syntactic and

semantic. What’s the difference between the two? This is

revision for you …


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Male: One comes from the grammatical form, from syntax, syntactic,

and semantic is from semantics, the meaning of the words, the

words themselves.

Lecturer: It’s true the semantic sequent has to do with meaning and

truth, and the syntactic sequent has to do with grammar and

structure, but one of them was defined in terms of – can you

tell me how a semantic sequent is defined, anyone?

Female: There is no logical something? (Laughter)

Lecturer: Almost. You're doing very well.

Female: No logical reason?

Lecturer: Not quite.

Male: Which one? Sorry, which one did you say?

Lecturer: Semantic we’re looking at.


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Lecturer: Start again.

Female: No way in which you can have something on the left that isn’t

true and still keep the other side the way it is? I don’t know …

Lecturer: You're in the right direction. The semantic sequent is defined

as ‘there is no logically possible situation in which the formulae

on the left-hand side of the sequent are all true and the

formulae on the right-hand side are false’.

The syntactic sequent, how is that defined?

It doesn’t take long, does it? It’s dissolved in a glass of wine.

(Laughter)

Female: Grammar and structure?

Lecturer: Something close … good.

Male: The set of things on the left … taken together with not the set

of things on the right is a closed thing.

Lecturer: Very good. Almost, yes. ‘The set consisting of the formulae on

the left ….’– except we don’t talk about that.

‘The set consisting of the counterexample of the argument is

closed’. That’s what it is. The counterexample consists of the

premises plus the negation of the conclusion. We will talk more

about counterexamples later.

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I think that’s quite enough revision. (Laughter)

(Slide 3) In this session we’re going to learn how to test

semantic sequents by using truth tables, whereas in the

session tomorrow we will learn how to test syntactic sequents

by means of truth trees, or ‘tableau’, they're called.

So how do we test semantic sequents by means of truth

tables?

Now, I'm being told by Chris not to stand in front of this,

because it makes me look ghastly. (Laughter) So obviously I'm

going to stay here the whole of the rest of the session.

(Slide 4) You’ve already met the truth table definitions of the

truth-functional connectives of propositional logic, so ‘P and Q’.

Remember the truth table – which you all came up with – I

didn’t have to tell you what it was … that’s what the truth table

is.

That gives you the definition of ‘and’, because it gives you the

truth value in every possible situation of ‘and’.

(Slide 5) Truth tables are tabular representations of all the

possible situations generated by the combination of truth

values.

You’ve got ‘P’ and ‘Q’. Each of them can be either true or

false. So there are four possible situations: four possible

combinations of truth value. Because these are truth-functional

connectives, each of the rows will have a truth value in it. That

truth value will give you the truth conditions for the whole

formula.

(Slide 6) Truth tables enable us to check whether it really is the

case that there’s no logically possible situation in which the

formulae on the left-hand side of the sequent are all true, and

the formulae on the right-hand side of the sequent are false.

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We use truth tables to check whether the argument claim –

the claim that the conclusion follows from the premises – is a

correct claim.

(Slide 7) To evaluate semantic sequents we need to have to

hand the truth table definitions of all the truth-functional

connectives, and luckily you have that on your handouts on

page 18 to 20.

(Slide 8) There they are again. What I've done here is I've put

them into one table, so ‘P and Q’ – whoa, I'm standing in front

of it, but never mind. (Laughter)

This one’s only got two in, because this is a unary connective.

So where ‘P’ is true, ‘not-P’ is false. Where ‘P’ is false, ‘not-P’

is true. ‘P and Q’, where they're both true, it’s true. Where

one’s false, it’s false. Where one’s false, it’s false, and so on.

That’s exactly what I told you earlier, except I've put them

together on one table, which makes it easier for me. But

you’ve got it on your handouts as well, because I won’t be able

to show you this when we’re working through it. You will need

to work from your handouts.

(Slide 9) So we’re going to learn how to evaluate semantic

sequents by evaluating this very simple sequent. You only

have to look at that to see whether it’s valid or not, don’t you?

Is it valid?

Male: Yes.

Lecturer: Yes, that’s valid. Because this means there is no logically

possible situation in which those are both true and that’s false,

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and you can see immediately that if those are both true then

that has to be true.

We’re working on a sequent where we know the answer to the

question ‘is it correct or not?’ So that’s quite useful.

First we’ve got to draw the empty truth table. To evaluate that

sequent you need a truth table. (Slide 10) You need to draw an

empty truth table, and you need to know how many columns

and how many rows to draw.

Now, you’ve seen me doing it, and none of you have

questioned how many rows and columns that I'm putting on,

and that’s because it is actually quite obvious.

For the number of columns you need a number of sentence

letters. How many sentence letters are there in that sequent?

Two – ‘P’ and ‘Q’. Plus the number of formulae in the sequent.

How many are there in that sequent? Three – one, two, three.

Two plus three is five. Plus a column for the sequent itself. So

we need six columns.

(Slide 11) Here you are – six columns. So you’ve got each

sentence letter, plus one formula, two formula, three formula,

plus one column for the sequent itself. Are you with me?

Male: Yes.

Lecturer: This is just the mechanical drawing of the truth table.

(Slide 12) To determine the number of rows we need to look at

the number of sentence letters.

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If there are two letters we need four rows. If there are three

letters we need eight rows. If there are four letters we need

sixteen rows and so on.

That’s because you're trying to show every possible

combination of the truth values of the letters.

So how many rows do we need for the sequent that we’ve got

here?

Male: Thirty two – no, sixty four.

Lecturer: No. You're jumping to conclusions here. We’ve got two

sentence letters, so how...?

Male: Four.

Lecturer: (Slide 13) We need four rows. That’s all. So we need a truth

table that – where’s the one that worked?

Male: There are a couple of them on the wall.

Lecturer: Good. I've got one here that works. So we need six across this

way, and two of them can be quite small, and then we need

one, two, three, four, five, six, and we need four across, like

that.

That’s because we need true, false, true, false, true, true,

false, false. That’s the way you always do it.

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The next one down is true, true, true, true, false, false, false,

false. Then you get all the combinations.

What do I put in here? ‘P arrow Q’, and then ‘P’, and then the

sequent itself, and then ‘Q’.

So that’s the empty truth table that will enable us to test that

sequent. Any questions from that?

Right, there you are. There’s the empty truth table. (Slide 14)

Now we’ve got to complete the truth table according to the

rules.

First we’re going to do the really easy columns. So, in a world

where ‘P’ is true and ‘Q’ is true, what’s the truth value of ‘Q’?

Female: True.

Lecturer: Dead easy, wasn’t it? In a world where ‘P’ is true and ‘Q’ is

false, what’s the truth value of ‘Q’?

Female: False.

Lecturer: You've all got this. What’s the next one?

Female: True.

Female: False.

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Lecturer: It’s exactly the same, in other words. All you’ve got is ‘Q’, so

you just transfer the truth values from ‘Q’ to that.

We can do the same with ‘P’. What do I put in here?


Lecturer: True, true, false, false. You’ve got the idea immediately. (Slide

15) So you see we’ve got here true, false, true, false, exactly

the same, and true, true, false, false. Anyone not with me on

that?

(Slide 16) Next we do the more complex formulae. We’ve got

the sequent left, but we will do that at the end. We can’t do that

until we’ve got to the end.

Here we’ve got a complex formula, and we need to fill that in

according to the truth table definition for conditional. So make

sure you’ve identified, on your truth table definitions, the

formula for the conditional. (Slide 17)

Female: I have a question about the formula for the conditional. Is it

supposed to be assumed at the moment or … [inaudible]?

Lecturer: Yes, you are just assuming it. That’s the one that you’ve got to

take on my authority for now, though you're quite right to

question it. It’s not obvious where it comes from at all, but

believe me, that’s it. We haven’t got time to …

Female: … I figured! [Laughter]

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Lecturer: We’ve got a whole session tomorrow for questions and

answers, so that would be a very good question to ask during

that.

So, given the truth table definition of the conditional, what do I

put in here? Can anyone tell me? What do I put here in row

one?

Male: True.

Lecturer: Does anyone not understand why I'm putting ‘true’ there?

Male: Should you not use small letters like the other ones?

Lecturer: Yes, I probably should. You're quite right. I don’t need to,

actually. Usually I say uppercase letters for whole formulae.

These are whole formulae. But I've started like that so I will

finish.

Does anyone not understand why I put ‘true’ there? No? Okay.

Look at the truth table definition of the arrow, which is this one,

this layer here. Where ‘P’ is true and ‘Q ‘is true, ‘P arrow Q’ is

...?

Female: True.

Lecturer: True. That’s why I put ‘true’ there.

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What do I put under row two, where I put ‘P is true’ and ‘Q is

false’?

Female: False.

Lecturer: False. What do I put under ‘P is false’ and ‘Q is true’?

Male: True.

Lecturer: True. And false, false...?

Female: True.

Lecturer: Is true. So it’s false, false, true. I'm just copying from the table.

There’s nothing odd about this.

The truth table for ‘P arrow Q’ tells us that that’s true unless

the antecedent is true and the consequent is false. I will justify

that tomorrow if you ask me in the question and answer

session.

(Slide 18) Does anyone not understand where I've got all

those truth values?

Where it would be harder is if this was ‘Q arrow P’. How would

I work it out if this was ‘Q arrow P’ instead of ‘P arrow Q’? Can

anyone tell me?

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Male: Just swap the two around? Swap P and Q in the truth table

that defines the arrow here …?

Lecturer: Yes. You’re absolutely right. What you do, so ‘P arrow Q’ is P

is here, true, and Q is true. It’s true. If it were ‘Q arrow P’, it

would actually be the same. You would have true and true. But

if it’s ‘Q arrow P’, instead of true, false, and therefore false, you

would have false, true, and therefore...?

Male: True.

Lecturer: True, yes. Do you see? So you have to be very careful, in

filling these in, that you’ve got the right sentence letter in the

right place.

This is an easy one, because I've done it in the same order as

the truth table definitions are giving you.

Let’s see where we are. Okay, so that’s where we are. We’ve

done the truth values for all the formulae now. (Slide 19) The

only thing we’ve got left to do is to check whether the sequent

itself is correct or incorrect.

So there we’re not looking at whether it’s true or false, we’re

looking at whether it’s correct or incorrect, because the

sequent says there is no logically possible situation – and each

of these is a logically possible situation, okay? (Slide 20)

There is no logically possible situation in which all the formulae

on the left-hand side are true and the formulae on the right-

hand side false.

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Well, is there a situation where all the formulae on the left-

hand side are true?

Female: Yes.

Lecturer: Which one?

Female: The top row.

Lecturer: (Slide 21) The top one, both of the formulae on the left-hand

side of the sequent are true, aren’t they?

Male: Yes.

Lecturer: Everything else is actually of no interest to us, because it’s not

the case that the formulae on the left-hand side are all true. So

it’s only that one that’s interesting.

Is it the case that all the formulae on the left-hand side are all

true and the formulae on the right-hand side are false, or not?

Female: No.

Lecturer: It’s not, is it? We’ll carry on.

(Slide 22) So we’re looking at number one. That’s the only one

that interests us.

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In row one, if the formula on the right-hand side is false then

the sequent is incorrect. (Slide 23) But it’s not false, is it? It’s

true. Therefore the sequent is correct, and we tick it, and we

see that that sequent is a correct sequent.

There’s no possible situation where the formulae on the left-

hand side are true and the formulae on the right-hand side are

false, therefore the sequent is correct, and we know that the

argument is valid.

(Slide 24) Would you like to test that sequent on your own?

Male: Just to check, you're not interested in any of the other results?

Lecturer: No.

Male: Because the other results don’t apply?

Lecturer: Because it’s only if all the formulae on the left-hand side are

true that we’re interested in the formulae on the right-hand

side.

None of these situations is such that all the formulae on the

left-hand side are true. One is true and one is false in each

case, and that’s therefore not interesting.

So we’re only interested in the ones where all the formulae on

the left-hand side are true.

So do that one on your own. Let’s see if you can do that on

your own.

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Male: That’s presumably why you rewrite the ‘P’ column?

Lecturer: Why we write the...?

Male: The ‘P’ column is repeated, but the second incarnation of the

‘P’ column is there to allow you to make the comparison easily,

without getting lost?

Lecturer: Yes.

Male: Suppose you just [inaudible]-

Lecturer: Because that’s a sentence letter and that’s a formula. That’s a

sentence letter and that’s a premise, if you like.

Male: Yes, okay.

Lecturer: That just gives us the key to the truth value, and that gives us

the actual truth value. They happen to be the same, because

they happen to be the same formula.

See if you can do that one. Put up your hands if you need my

help.

(Slide 25) Just draw the truth table for now and don’t start

completing it. I mean complete the truth value to the sentence

letters, but don’t do the formula yet.

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Put up your hands if you want my help.

So how many columns do I need?

Male: The same number as last time.

Female: Six.

Lecturer: Six? Okay, so one, two, three, four, five, six.

How many rows do I need … [inaudible discussion]? How

many rows do I need [apart from that]?

Female: [Four].

Lecturer: Four? Okay, one, two, three, four. What do I put in here? P, Q.

Then the truth values down here?

Male: TTFF.

Lecturer: TTFF and TFTF, good. Then what do I put here?

Female: ‘P then Q’.

Lecturer: ‘P then Q’.

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Male: [inaudible discussion]. Oh, sorry.

Female: Oh, sorry.

Lecturer: No. All we’re doing at the moment is the truth table, the empty

truth table.

Male: ‘Q’.

Lecturer: ‘Q’.

Female: Sequent.

Lecturer: Sequent.

Male: ‘P’.

Lecturer: ‘P’. Good. (Slide 26) So that’s the truth table. That’s the empty

truth table. We haven’t started proving anything yet.

Is that what people had?

Male: Can you put an F on the bottom row of Q, TFTF?

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Lecturer: Sorry. You're quite right. Good.

(Slide 27) The next thing we do is we start proving it. The truth

table should be very like the one we drew for the last sequent.

In fact, it’s identical to the one we drew for the last sequent,

except it’s got a different sequent in it.

So your next steps, put in the appropriate truth values for the

simple formulae, the complex formulae, and then for the

sequent.

Are we ready? Would people like more time or should we get

on?

What do I put under here?

Male: TTFF?

Lecturer: No, Chris, you know how to do this. Let’s have someone else.

What do I put under the ‘P’?

Female: TFF.

Lecturer: TTFF, exactly the same as under ‘P’ there, exactly the same.

What do I put under the ‘Q’?

Male: [The same].

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Male: TFTF.

Lecturer: TFTF, because it’s exactly the same there, that is just ‘Q’. So

what’s the truth value of ‘Q’ in the world in which ‘Q ‘is true?

It’s true, obviously. What goes in here?

Male: TFTT.

Lecturer: Who said that? Steve? TFTT.

Small cases again, but never mind. It doesn’t matter.

That comes from the truth table definition of the conditional

again.

Now we get to check it. Looking at each of these rows

separately, because each row represents a different possible

world, is there a possible world in which the premises are all

true – in which the formulae on the left-hand side of the

sequent are all true?

Female: The first one.

Lecturer: The first one, so that one. Is there any other one?

Male: Yes.

Lecturer: Which one?

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Male: The third.

Lecturer: Three is also such that both the – sorry, it’s not obvious, but

that’s because of my writing …

But ‘if P then Q’ is true, because ‘P’ is false and ‘Q’ is true,

therefore it’s true, ‘Q’ is true, therefore that one, the formulae

on the left-hand side, are also all true.

(Slide 28) Now we need to check whether there’s a situation in

which they are all true on the left-hand side and yet false on

the right-hand side. Is there one?

Male: Yes [inaudible].

Lecturer: Well, that’s not one, is it, because that’s true on the right-hand

side as well? But that one is a counterexample.

Male: Sorry, I've missed something here. I thought ‘P’ was false on

number three?

Lecturer: You thought was ‘P’ was false on number three? It is.

Male: Yes.

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Female: I made exactly the same thing. You’re not supposed to count

when you're looking at it. You just count the formulae bit. So

this bit, ‘if P then Q – T’, and then the ‘Q’, and you don’t count

the ‘P’ and ‘Q’ initially.

Lecturer: These are just the key. These two both tell you only what the

truth values are. They're telling you all the different

combinations.

Given that you’ve got two truth values, you need a world in

which both are true, both are false, and one of them is true,

one of them is false. That’s the same on all of them.

In the world in which ‘P’ is false, then ‘P’ is false. So the

formulae on the right-hand side is also false.

Here you’ve got a situation where both the premises are true,

and the conclusion is false, and therefore that’s a

counterexample to the argument.

Does that make sense? This ‘P’ is very different. This ‘P’ (the

one on the RHS of the sequent) is a conclusion, and this ‘P’

(the one in the key) is actually just the key.

This ‘P’ is just telling us the different truth values, in order to

give us the world that we’re looking at, in order to determine

whether the sequent is true or false in that world.

So all we want is one possible world in which all the premises

are true, and the conclusions false, (Slide 29) and we know

that that sequent is incorrect, and therefore …

Male: So it means the whole sequent is incorrect?

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Lecturer: It means the whole sequent is incorrect.

Male: So you could have three true, three-

Lecturer: Remember the definition of validity is there is no possible

world in which all the premises are true and the conclusion

false.

These are all the possible worlds. So what we’re saying is

there is no possible world.

So one world in which the premises are true, and the

conclusion false, is enough to show that the whole thing is

invalid.

Male: So if you had a table of 64 entries, and you only one cross

somewhere-

Lecturer: One cross shows you the sequent is incorrect, the argument is

invalid, yes.

Male: Ah, right.

Lecturer: It doesn’t matter how many ticks you get – unless all of them

are ticks, in which case you know it’s correct.

Male: You can ignore the ones of which the other side is ‘both not

true’?

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Lecturer: Anywhere it’s not the case that all the left-hand side are true,

that’s not of interest, because you haven’t got a situation

where the premises are all true.

An argument is only good if the premises are all true and it’s

valid, i.e. it can’t be the case that the premises are true and the

conclusion false.

Male: However complicated, one cross means the whole argument

fails?

Lecturer: However complicated, one cross means the whole argument

fails.

Male: I think earlier you mentioned ‘affirming the consequent’?

Lecturer: The reason this is invalid is it’s affirming the consequent, yes.

This is a very typical fallacy. It’s a very common fallacy. It is a

fallacy because it’s affirming the consequent, yes.

Male: We’ve only been dealing with ‘P’ and ‘Q’. If you go [beyond the

boundary and ‘P’, ‘Q’, ‘R’ and whatever, would it still work?

Lecturer: Oh, yes, it will work. It doesn’t matter how many sentence

letters you have, this will always work. It always works. You

won’t ever be left with uncertainty on this. You will always get

an answer. Let’s -

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Male: Sorry, I've just realised what the fallacy is now you've

explained it. ‘P therefore Q’, ‘P’ so ‘Q’, this is ‘if P then Q’, ‘Q’

then ‘P’ …

Lecturer: Yes, except you shouldn’t be interpreting that [the sequent] as

‘then’. This is not an ‘if/then’.

Male: No.

Lecturer: There’s a big difference between implication and entailment.

So there you are. (Slide 28 again) This is exactly what we’ve

just done. In row one, the formulae to the left-hand side are all

true, but so is that to the right, so that’s not a counterexample.

But in row three, the formulae to the left are all true, but the

formula to the right is false, and that’s exactly what we need

for a counterexample. That’s enough to tell us that the whole

sequent is incorrect.

Male: So it doesn’t matter, in this context, if there is no situation in

which both propositions are true? If you constructed a sequent

and you built your table, and there were no examples of where

all of the propositions were true on the left-hand side, that

wouldn’t matter?

Lecturer: No.

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Male: It doesn’t matter that there are no ticks. It only matters if

there’s one cross.

Lecturer: The crosses are the only things that are really interesting. We

only put the tick in because you see ‘two trues’. All the

premises to the left-hand side are true.

Male: If there are no ticks and no crosses, what then?

Male: It’s still a valid argument…

Lecturer: It might be all like that. That’s fine.

Male: Is it a valid argument?

Lecturer: It’s not invalid. (Laughter) Given that it’s either valid or invalid,

it not being invalid means it’s valid, yes.

Male: Is this the root of the reason why a lot of science is done by

disproving stuff rather than proving stuff?

Lecturer: Yes.

Male: Because this process can’t prove, but it can invalidate.

27

Lecturer: The reason that Popper was so anti-induction is because

induction never gives us certainty. He thought that what we

need to do is falsify arguments instead of proving them,

because you can’t prove anything, but what you can do is

falsify them. You can deductively falsify things.

Male: Yes, which is what we’ve just done, but that’s the basis of -

Lecturer: Popper’s …

Male: That’s the basis of the null hypothesis test, with clinical trials in

epidemiology. You always prove that the null hypothesis is

wrong. You never prove your hypothesis is right.

Lecturer: Yes. You shouldn’t be trying to confirm your thesis. You should

be trying to ‘disconfirm’ it. You should be trying to falsify it.

Male: Yes.

Lecturer: (Slide 29 again) Good. So we now know that the sequent ‘P

arrow Q’, ‘Q’ – you see, I'm doing it now – ‘sequent’ ‘P’ is

incorrect, and the argument therefore invalid.

The interesting thing is we’ve also got the counterexample to

the argument. We can say that this argument is invalid

whenever ‘P’ is false and ‘Q’ is true.

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If we had an interpretation, which we don’t, because we’ve just

been working with the sentence letters, but if we had an

interpretation we can get exactly what the counterexample is.

So we could see under what conditions this argument is

invalid.

So not only do you know with complete certainty that it is

invalid, you also know when it is invalid, what the situation is

that makes it invalid.

Well, so far we’ve done baby steps, and these are really easy

arguments. We knew before we even started whether they

were invalid or not.

(Slide 30) Now let’s do this one. So this is the sequent. Do you

remember what that’s about? Does anyone remember what it’s

about?

Female: Tickling cats.

Lecturer: Tickling cats, exactly so. That’s the formalisation of the ‘If you

tickle her you deserve to get scratched’ type of argument. If

you remember, that was the interpretation. (Slide 31) It’s all

about whether she wanted you to tickle her, and whether you

went about it in the right way, and so on.

To test this sequent, naturally we first draw the truth table.

Okay, draw the truth table. (Slide 32)

Female: Are you telling us to?

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Lecturer: Yes. (Laughter) Well, you know the algorithm … So you need

to work out how many columns and how many rows.

Female: Can we have it up again, because I can’t remember it.

Lecturer: Yes, sorry. There’s the formula. Has anyone worked out how

many columns we’ve got yet?

Male: Yes.

Lecturer: Put up your hand if you have. Well done. How many?

Female: Ten.

Lecturer: Ten? Is that what everybody has got?

Male: Nine.

Lecturer: Some of you think ten, some of you think nine. I think it’s nine,

actually.

Male: Nine, yes.

Lecturer: You're getting an extra one from somewhere.

30

Male: Yes, I'm going with ‘not P’ column, separate to the …

Lecturer: No, that’s a mistake.

Male: Oh.

Lecturer: Well, where would you have a ‘not P’ column?

Male: [inaudible discussion]

Lecturer: Well, then you shouldn’t have that, should you?

Male: Oh, okay.

Lecturer: Yes. There are nine columns. Okay, good, nine columns. Now

I want to know how many rows there are.

Male: Sixteen.

Lecturer: Sixteen? Well done. You’re getting this very fast. Well done.

Female: I have to make this, I tell you now.

31

Lecturer: So here we are. There we are. (Slide 33) Nine columns and

sixteen...

The only reason we work out sixteen is we’ve got ‘true false,

true true false false, true true true true false false false false,

true true true true true true true true, false false false’...

You can see why truth tables are a bit of a bore, and why you

will be very glad to get onto tableau tomorrow, which are

much, much easier!

Tableau work just as well as truth tables, but they're much,

much easier, but only after you’ve learnt how to do them. Truth

tables …

Female: Can’t we do those first?

Lecturer: No. (Laughter)


Lecturer: How many columns there are? Well, I gave you the formula

earlier. If you’ve got two sentence letters there are four. If

you’ve got three sentence letters there are eight. If you’ve got

four sentence letters there are sixteen.


32

Male: It’s two raised to the power of the number of letters.

Male: Yes: two squared by nine, four squared.

Lecturer: There you are.

Male: No, it’s not squared, because you’ve got …

Male: No, sixteen is three squared by four squared. It’s a nine to

sixteen matrix, isn’t it?

Lecturer: Ask him, not me. (Laughter)

Male: This is ....

Male: In the first column the convention to have all Ts and then Fs is

obvious. In the second, third, and fourth column is there

anything simple to remember the convention, or do you just …

Lecturer: What you’ve got to have is the situation for all the possibilities,

and this is just a way of making sure you’ve got that. So this is

‘true true true true, true true true, false, true true true, false,

true true, false false’, etc.

33

Male: So, if we were doing it on our own, certainly as long as we

have all the combinations we’d have it, but this is the

convention.

Lecturer: This is the way we are doing it, yes.

Male: Right. Because you could have all Ts in the second row …

Lecturer: Well, this is the way to remember it. It’s knee-jerk stuff. It’s

dead easy to do this once you’ve started to do it.

Now we need to start doing the formulae. (Slide 34) Now,

every one we’ve done so far, we’ve done complex formulae

but we haven’t done one with two truth-functors in. Here we’ve

got one formula that has two truth-functors. Which one is it?

Female: The last one.

Lecturer: It’s the last one. It’s the one on the right-hand side. You’ve got

a negation sign and a conditional.

What are we going to do first, do you think: the negation sign

or the conditional?

Male: Well, the negation sign only applies to ‘P’ in this case.

Lecturer: Good, yes. We do the truth-functor with the smallest scope first

when we’re doing truth tables.

34

The one with the smallest scope, we’ve got the ‘not-P’. The

scope of the ‘not’ is just ‘P’. If the ‘not’ were outside the

brackets, the scope of the ‘not’ would be ‘P arrow S’, but here

it’s just ‘P’.

So what we’re going to do is, in lowercase letters, we want to

put in the truth value of ‘P’. And how are we going to do that?

Male: Just reverse the first P?

Lecturer: It’s just going to reverse whatever we’d had in ‘P’. So it’s the

‘P’ that’s negated, so this is going to reverse whichever truth

value ‘P’ has in the key. (Slide 35) So there we are. ‘P’ is true,

‘not-P’ is false. ‘P’ is false, ‘not-P’ is true, and so on. Do you

see how that works?

So having done the ‘not-P’, we now want to put in the

conditional. This is where I warned you before: you’ve got to

be very careful about using the truth table definition of the

conditional, because this is the antecedent: not ‘P’. Are you

with me?

Male: Yes.

Lecturer: So here we’ve got ‘S’. (Slide 36) Well, we could actually put in

the truth – I think I've done that, actually. Let me see whether I

have. Yes, I have. I've put in the ‘S’ here. (Slide 37)

You will notice that this is exactly what’s there has gone in

here. That’s just to make it much easier for us to fill in the truth

35

value of the whole formula here, because we’ve actually go the

truth values on each side.

As you get more practiced you don’t have to do this. You can

just do it from the key. But let’s not do that. Let’s not run before

we can walk. So what do we put: false, true? What’s the truth

value for the conditional? (Slide 39)

Chris: Whatever it is.

Lecturer: True or false? ‘Whatever it is’, says Chris. (Laughter)

Male: True.

Lecturer: That’s true. Okay, what’s ‘false, false’?

Male: True.

Lecturer: ‘False, true’?

Male: True.

Lecturer: Actually, you will notice that every time the antecedent of a

conditional is false, the conditional is...?

36

Male: True.

Lecturer: (Slide 39) True. So we can put ‘true’ in all of those. Are you

with me?

If you look at the truth table definition for the conditional, you

will see that whenever the antecedent is false the conditional is

true. So, as the antecedent is false in all these, all of these will

be true. So actually we can just put ‘T’ down all those and start

looking here.

When the antecedent is true, and the conditional is false, the

truth value is...?

Male: True.

Lecturer: True. When the antecedent is true and the conclusion is false,

false. So we’ve got our first ‘false’ here. True or false?

Male: True.

Lecturer: True. True or false?

Male: False.

Lecturer: False.

37

Male: True.

Lecturer: True, false, true, false. Are you with me?

Male: Yes.

Lecturer: So there we go. Have a look at that and see if there’s anything

you don’t understand. Put your hands up if there is.

Next we’re going to do this one. Looking at your truth table

definition for ‘or’, see how to fill those in.

That one’s quite easy, because you’ve got ‘PQ’ in the right

order there, so actually that’s quite a nice one to do.

Let’s move on. There’s one. (Slide 40) That’s the truth table

definition for disjunction – the ‘or’ – and that tells us that it’s

only false when ‘P’ is false and ‘Q’ is false.

So when we fill that in here, (Slide 41) if you look at ‘P’ and ‘Q’

here, we’re looking for when they're both false, and actually

that doesn’t happen until down here, and there are four rows in

which they are both false, and in each case each of those is

false, otherwise they're true. Are you with me?

Female: Yes.

Lecturer: So ‘P’ or ‘Q’ is true, then ‘P’ and ‘Q ‘are both true. When ‘P’ is

true and ‘Q’ is false, ‘P or Q’ is true. The only time where ‘P or

Q’ is false is where ‘P’ is false and ‘Q’ is false. Are you with

me?

38

Male: Yes.

Lecturer: So that one’s filled in. Again, it just comes mechanically from

the truth table definition of ‘or’.

Now we need to do the two conditionals. (Slide 42) The

hardest thing about these is getting the – in each case they are

in the same order – but it’s getting the truth values in the right

place from the key.

Okay, ‘Q arrow R’. Tell me why that’s true, somebody. Anne,

can you tell me – sorry, picking out Anne is so mean, but go

on. Can you tell me why that’s true?

Anne: Because ‘S’ is true.

Male: Which one is this?

Lecturer: That’s not true, because ‘S’ is true. That’s not the only reason.

Male: It is.

Lecturer: Well, yes, but it’s not from what I'm … because ‘R’ is false and

‘S’ is true. So that gives you true here, because that’s what,

again, the truth table definition says.

Why is that false? Can somebody tell me that?

39

Male: ‘P’ is false.

Lecturer: Because ‘R’ is true and ‘S’ is false, and therefore ‘if R then S’

is false. That’s the only time when ‘if R then S’ is false, is when

the antecedent is true and the consequent is false.

(Slide 43) Does everyone see where these are coming from?

Or if they don’t see particular ones, they’ve got the general

idea?

Male: You say, ‘Do you understand it?’ I say, ‘I can fill the table in,

but I'm not sure I actually understand what it means.’

Lecturer: Being able to fill the table in is quite enough at the moment.

Well, you do understand what it means, because what you're

doing, we know that each one of these truth-functors is truth

functional. So the truth value of the whole depends only upon

the value of its parts.

Male: Yes, I understand that.

Lecturer: All we’re doing is saying what the impact of the combination of

the truth value of the parts is.

Male: Yes.

40

Lecturer: We know how to work that out from the truth table definitions.

(Slide 44) Now we can fill in this column, which will tell us

whether the argument is valid or not, or whether the argument

sequent is correct. So is there a possible situation there in

which all the premises are true?

Female: Yes.

Lecturer: So give me the numbers of the rows. So number one, row one

is true true true. Good. Two is true false, so it doesn’t count.

So the next one is this one, isn’t it? True true true. This one is

true true true. This one and this one … So quite a few of them,

actually.

Male: There are five.

Lecturer: Here they are. (Slide 45) There we are. There are lots of them

in which the – well, I've filled in the tick already, so no.

We now need to check each of these to see whether we’ve got

a false on this side, and wherever all the premises are true the

conclusion is also true, isn’t it?

Male: Yes.

Lecturer: So there isn’t a counterexample. There is no possible situation

in which the formulae on the left-hand side are all true and the

41

formulae on the right-hand side are false. Therefore the

sequent is correct. (Slide 46)

Male: Would I be safe in only considering the ones that are false on

the right-hand side and not being interested in the rest?

Because to find the counterexample …

Lecturer: Well, that’s not a counterexample, is it?

Male: No, but I'm only going to look at the ones that say F on the

right-hand side.

Lecturer: Yes.

Male: I don’t need to discuss any of the ones that say ‘T’.

Lecturer: Well, it doesn’t matter which way round you go. You have to

look at whether what’s on the left-hand side is true and what

on the right-hand side is false. You can start by looking at

what’s false on the right-hand side.

Male: Just that there are fewer of them …

Lecturer: Yes, it doesn’t matter at all.

42

Male: There are only four places where the sequent comes out as an

F.

Lecturer: One, two, three, four.

Male: So you only need to look at those rows.

Lecturer: You could look there.

Lecturer: Yes.

Male: But they’ve all got an ‘F’ in them, so you can...

Male: So there’s no counterexample?

Lecturer: Yes, there’s no counterexample. So that argument claim is

correct. There is no possible situation in which the formulae to

the left-hand side are true and the formulae to the right-hand

side are false. Therefore the sequent is correct, and we know

that the argument is...?

Male: Valid.

Lecturer: Valid, yes. So the sequent is correct, the argument is valid.

(Slide 47) This is the argument with which we started, and we

43

know for absolutely certain, completely conclusive, that this

argument is valid.

There is no possible situation in which the premises of that

argument are true and the conclusion of that argument is false.

So it’s a good argument. (Slide 48) There are some semantic

sequents for you to practice on.

Male: Thank goodness! (Laughter)

Lecturer: And incidentally, what do you think this means?

Male: Always true?

Lecturer: There is no possible situation in which what’s on the left-hand

side is true, is all true.

So what that’s telling you is that that (the formula) is

inconsistent. There is no possible situation in which the

formulae on the left-hand side are all true. So in other words

that must be inconsistent. There is no formula on the right-

hand side, so you can ignore it.

This one tells us what? That says that that’s (the formula) a

contradiction, so in every possible world that’s false. That tells

us what?

Male: It’s always true.

44

Lecturer: That this is a necessary truth. So there’s no possible situation

in which the formula on the right-hand side is false. That’s

always true, in other words. So you can read it in exactly the

same way. You just haven’t got one of the sides to worry

about.

Male: It says answers in your answer booklet on page seven. Are we

still waiting for the answer booklets?

Lecturer: You are, yes. I've got the answer booklets here. I will give

them out. You didn’t think I was going to give them to you.

(Laughter)

I'm a bit unsure about giving them to you tonight, because you

might look up tomorrow’s answers. The only thing is that I'm

absolutely confident that, if you do look up tomorrow’s answers

tonight, they won’t mean anything to you. (Laughter)

Okay. We’ve actually got five minutes for questions.

Male: Can I just ask a question?

Lecturer: Yes.

Male: I'm very bothered about the rows that you just put dashes in,

how they don’t seem to contribute at all to the validity, or

otherwise, of the argument.

45

Lecturer: There are two questions to ask of an argument. Are all the

premises true? Is the argument valid?

Male: Yes.

Lecturer: If it’s not the case that all the premises are true, then actually

you're not interested.

Male: Oh, okay, yes.

Lecturer: Let me be careful about that. In a truth table you’ve got all the

possible combinations, and that’s the important thing.

Male: Of ‘P’, ‘Q’ and ‘R’, or whatever your …

Lecturer: Or of whatever your premises are.

Male: But then you’ve got to have all the arguments true?

Lecturer: Not arguments. All the formulae on the left-hand side, i.e. all

the premises, have got to be true, before we are interested in

the truth-value of the conclusion.

Male: If they're not true that contributes towards a dash?

46

Lecturer: Yes. Well, that is a dash.

Male: That is a dash?

Lecturer: Yes. So if some of the premises are true, some are false

[cross-talking] …

Male: Well, then you can’t tell anything about the argument,

basically?

Lecturer: Yes.

Male: Okay.

Female: On page four, the bit at the bottom, where it talks about ‘P’

being false and ‘Q’ being true.

Lecturer: Page four?

Female: Yes. But then ‘if P then Q’ is true. Is there somewhere I can

get more information on how that’s logically derived? Because

I kind of struggle with that intuitively.

47

Lecturer: Looking at four.

Female: Page four. It’s the definitions, I think.

Male: It’s what we discussed earlier.

Male: The bottom of page four.

Male: ‘If P then Q’ where ‘P’ is false and ‘Q’ is true.

Lecturer: Are you talking about slide fifteen?

Female: Slide eight.

Lecturer: Eight?

Female: I think it’s slide eight.

Lecturer: So it’s not page four.

Male: We’ve got two per side.

Lecturer: Oh, yes, I'm sorry. Give me one with two per side.

48

Male: The slide numbers …

Lecturer: Oh, right, the definitions. Sorry, go on. What was the question

again?

Female: I'm just struggling a bit with ‘P’ being false and ‘Q’ being true,

and then ‘if P then Q’ being true as a result. So I was just

wondering if there’s anywhere I can look up more about the

logic behind that.

Lecturer: That’s the one that Kirsten is going to ask me about in the

question and answer time tomorrow. Well, we could do it very

quickly now.

Well, no, I think it’s probably better to do it tomorrow in the

morning. Because that is very difficult to motivate, and I can

quite see why you're having trouble with it. I will try and

motivate it for you tomorrow.

Female: Thank you.

Male: A few moments ago, when you mentioned Popper, in your

example previously, where there’s no counterexample, in his

language that would be not falsifying it?

49

Lecturer: No, it’s ‘not falsified’.

Male: Not falsified?

Lecturer: Yes. Think of it like this. Think of the logic of investigation, if

you like.

You start here with sightings of white swans. By inductive logic

you start to say, ‘All swans are white.’ Every swan I've ever

seen has been white, therefore by an induction, possibly a

rather bold one, I form a hypothesis that all swans are white.

Then I test that hypothesis by saying, ‘well, if all swans are

white, then anything that is a swan will be white’. Well, of

course that’s not going to be true. If I see a black swan, that’s

falsified that immediately. That’s by deductive logic. So

deduction falsifies.

So every white swan I see is not very interesting, because it’s

just yet another little confirmation of my hypothesis, for which I

already had – every swan I've ever seen has been white. ‘Oh,

look, there’s another one, and another one, and another.

Yawn. (Laughter) There’s a green one. Now we’re talking.’

This is really interesting, because once you’ve got a green one

you know that that (the hypothesis) is wrong.

Female: You only need one of them.

Lecturer: You only need one of those and you’ve completely

conclusively falsified that.

50

Male: By knocking out the false one …?

Lecturer: Actually, you haven’t quite, because the other thing. There’s

always some interpretation. Because of course you might see

a black swan going up the Swan River, in Perth, and you think,

‘that can’t be a swan. It’s not white.’ (Laughter)

Male: You could look at it.

Lecturer: Yes. ‘It looks like a swan, but it’s obviously not one, because

it’s not white, and if all swans are white...’

Another one that goes like that is, ‘All women are passive. Mrs.

Thatcher isn’t passive. Therefore Mrs. Thatcher ... Well, either

you will falsify your premise and say, ‘It’s not the case that all

women are passive’, or, ‘Mrs. Thatcher isn’t a woman.’ That’s

where the joke ‘Mrs. Thatcher is the best man in the Cabinet’

[came from]. (Laughter)

Male: False negatives or false positives. Because a black swan that

you see might be a very dirty white swan!

Lecturer: Well, but then it’s the premise isn’t true, is it?

Male: Well, you think you’ve found the exception that allows you to

dispense with the inductive rule that all swans are white.

51

Lecturer: No, that’s an – sorry, inductive, you said that, yes.

Male: You think you've found the overturn of that rule, but in fact you

have made an error.

Lecturer: But remember that the deduction tells you that if the premises

are true the conclusion must be true. What you're describing is

a situation where one of the premises isn’t true. It’s not the

case that this swan is black. It’s a white swan that’s dirty.

Male: But you do get false positives and false negatives …

Lecturer: Well, yes, but this doesn’t change the fact that if the premises

are true the conclusion must be true, and that’s what deduction

gives you.

On one side you’ve got induction. You form your hypothesis on

the basis of inductive evidence. Then you use deductive logic

to test ...

Male: To disprove it?

Female: Is there some way of ‘weighting’, where if it’s inductive, if you

have seen 50 swans …

Lecturer: Oh, you have to.

52

Female: … And you have another person who has seen 60 swans that

– do you know what I mean? It progresses. It becomes a

statement that is now worth 60 swans.

Lecturer: Well, the reason that science is a collaborative enterprise is

that you're always wanting people to confirm your – you don’t

know whether it’s just happening in your lab, whether in

Australia things are completely different, as indeed they were

with swans.

That’s why you want other people to replicate whatever it is

you’ve concluded, because each replication is further

confirmation.

But of course it doesn’t matter how often you confirm the thing,

you’ve never got certainty, because you never get certainty

with an inductive argument.

Female: You get increasing …

Lecturer: … you falsify. Well, actually, remind me. I will talk to you, if you

like, if you ask me tomorrow, which would be very nice of you,

about the paradoxes of confirmation. I will talk to you about

black ravens, and white gym shoes, and things like that, and

about grue. That will be fun …

Male: This is why scientists eventually stop doing experiments. You

see one white swan …

53

Male: No.

Male: No, the way you do an experiment is you – the way you do

epidemiology, in clinical trials, is you turn that on its head, and

then seek to...

Lecturer: Well, you do exactly what I...

Male: The epidemiologist would not phrase the question that way. I

think they would phrase the question: ‘There are some swans

that are not white.’ Then they would look at evidence to

discard that hypothesis. They ask the question a very odd way.

It takes a long time to get your head around the way they...

I may be mistranslating this. If you set out to say, ‘My

treatment is better than the gold standard of care’, that’s not

the question you ask. That’s what you want, but that’s not the

question you ask. The question you ask is, ‘Can I find

evidence to reject the idea that my treatment is equivalent to

the gold standard?’

Lecturer: Yes, but that’s exactly what I'm saying.

Male: Which is how they set out to do their experiments.

Lecturer: That’s exactly trying to falsify your hypothesis, which is that

your treatment …

54

Male: Is the same as.

Lecturer: Either the same as, or better, or whatever …

Male: They always set out to falsify the idea that the two are

equivalent.

Female: So they're postulating?

Male: If I'm getting this right, they’re falsifying the opposite of your

hypothesis. Is that what you're saying?

Male: Yes, I think so. I have a bright idea that I can cure cancer, and

the gold standard of care is radiotherapy. So I'm claiming that

my idea is better than radiotherapy.

The way I would phrase the clinical trial is, ‘Radiotherapy is

equivalent to my bright, new idea.’ So I would phrase the

hypothesis, ‘My good idea is the same as radiotherapy’, and

then I would look for statistical evidence to reject that idea, or

fail to reject that idea.

Lecturer: So you would design an experiment or a trial which would

show that P is not equivalent. Actually, you don’t want

equivalent. Radiotherapy is worse than P.

Male: I want to reject the idea the two are equivalent.

55

Male: But you design it …

Lecturer: So you can either show that P is better or P is worse? Would

either …

Male: Well, yes, there are statistical tests. There are two different

types of statistical test: one of which will show you which way,

and the other which just says, ‘They're not equivalent.’

Lecturer: Sadly, as nobody ever publishes negative results, we will

never actually know whether you succeeded in showing this.

Male: That depends whether you work for the university or

[inaudible]. (Laughter)

Male: A physicist would publish the negative results. That’s all

they're interested in doing is publishing …

Lecturer: Well, they’d get it put in a journal.

Male: When they discovered that neutrinos went faster than the

speed of light, that got published, whereas ...

Lecturer: Well, yes, but...

56

Male: It was completely wrong, but when they were promoting …

Lecturer: And took a long, long time before they agreed that they had

published it and they probably shouldn’t have done.

Male: Oh, yes. That’s also true …

Lecturer: Well, they wanted to get everybody working on what they’d

done wrong.

Male: Well, they just wanted to know what they had done wrong.

They knew they had got something wrong, but they had no

idea what it was.

Lecturer: Let’s go, because it’s well after time, and I need to go home.

(Slide 49).

Documents

A romp through the foothills of logic Session 3 › 2014 › 07 › fl3.pdf · Firstwe’vegottodrawthe emptytruth table.Toevaluatethat sequent you need a truth table. (Slide 10)